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We introduce a model for Non-Standard neutral current Interaction (NSI) between neutrinos and the matter fields, with an arbitrary coupling to the up and down quarks. The model is based on a new $U(1)$ gauge symmetry with a light gauge boson that mixes with the photon. We show that the couplings to the $u$ and $d$ quarks can have a ratio such that the contribution from NSI to the Coherent Elastic Neutrino-Nucleus Scattering (CE$\nu$NS) amplitude vanishes, relaxing the bound on the NSI from the CE$\nu$NS experiments.
Additionally, the deviation of the measured value of the anomalous magnetic dipole moment of the muon from the standard-model prediction can be fitted.
We discuss the relevant experimental and observational bounds on the parameter space and show that there are ranges that can lead to both a solution to the $(g-2)_\mu$ anomaly and values of $\epsilon_{\mu \mu} = \epsilon_{\tau \tau}$ large enough to be probed by future solar neutrino experiments.
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